Explain precision/recall and compute NN output

You are given a short ML fundamentals assessment with three parts.

Part A — Precision/Recall/F1

A binary classifier on a dataset produced the following confusion-matrix counts:

• True Positives (TP) = 40
• False Positives (FP) = 10
• False Negatives (FN) = 20
• True Negatives (TN) = 130

1. Compute precision , recall , and F1 .
2. If you raise the decision threshold , what typically happens to precision and recall (and why)?
3. In an imbalanced dataset where positives are rare, when would you optimize for precision vs. recall?

Part B — Ensemble learning (select all correct)

For each statement, mark whether it is generally True or False, and briefly justify.

1. Bagging primarily reduces variance .
2. Bagging uses bootstrap sampling (sampling with replacement) to train each base model.
3. Boosting trains base learners independently and can be fully parallelized without changing the algorithm.
4. Ensembles can improve generalization because averaging/voting can cancel out uncorrelated errors.
5. If the base learners are perfectly correlated (always make the same predictions), bagging will provide large gains.

Part C — Forward pass of a small neural network

Compute the network output for the following fully connected network. Use the sigmoid activation $\sigma(t)=\frac{1}{1+e^{-t}}$ at both layers.

• Input $x = [1.0,\ 2.0]^T$
• Hidden layer (2 units): $h = \sigma(W_1 x + b_1)$
  • $W_1 = \begin{bmatrix}0.5 & -1.0\\ 1.0 & 0.5\end{bmatrix}$ , $b_1 = \begin{bmatrix}0.0\\ -0.5\end{bmatrix}$
• Output layer (1 unit): $\hat{y} = \sigma(W_2 h + b_2)$
  • $W_2 = \begin{bmatrix}1.5 & -2.0\end{bmatrix}$ , $b_2 = 0.1$

Return $\hat{y}$ as a decimal rounded to 4 digits.